It is customary to adorn our homes and synagogues on Shabuot with flowers and greenery. This custom is recorded by the Rama (Rabbi Moshe Isserles of Cracow, 1525-1572) in his discussion of the laws of Shabout (Orah Haim 494:3). More so than on other holidays, it is appropriate on Shabuot to decorate the homes and synagogues with flowers and other vegetation, in order commemorate Matan Torah. At the time of the Revelation at Sinai, G-d warned the people not to allow their cattle to graze on the mountain ("Gam Ha’son Ve’ha’bakar Al Yir’u El Mul Ha’har Ha’hu" – Shemot 34:3), indicating that it was full of pasture. Mount Sinai is in the desert, where vegetation does not grow, but in honor of the event of Matan Torah G-d had flowers and grass grow on the mountain, and we commemorate this adorning of Mount Sinai by adorning our homes and synagogues on Shabuot.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) adds a deeper interpretation of this custom based on Kabbalistic teachings. He explains that Torah study has the effect of "Mituk Ha’dinim" – "sweetening" harsh judgments issued against a person. Even if G-d had issued a decree, Heaven forbid, against somebody, that decree can be annulled through the study of Torah. The divine Name associated with strict judgment is "Elokim," whereas the Name associated with "Mituk Ha’dinim" is "Havaya" (Y-H-V-H). When we spell out the names of the letters of these Names (e.g. the Alef of "Elokim" is spelled out, "Alef, Lamed, Peh"), the combined numerical value of the letters of "Elokim" is 300, and the combined numerical value of the letters of "Havaya" is 72. The letter representing the number 300 is "Seen," and the number 72 is represented by the letters "Ayin" (70) and "Bet" (2). The two letters of "Havaya," the Ben Ish Hai explains, surround the letter of "Elokim" to neutralize its effect, such that the letter "Seen" is placed in between the "Ayin" and "Bet." When the three letters are arranged in this fashion, they spell the word "Eseb" – "grass." Thus, we put out vegetation on Shabuot to symbolize the profound spiritual effect of our Torah study, how it has the capacity to annul harsh judgments. As we celebrate the event of Matan Torah, we remind ourselves of how valuable and precious Torah learning is, as alluded to in the "Eseb" with which we decorate the synagogue and home.
Summary: It is customary to decorate homes and synagogues with vegetation on Shabuot to commemorate the vegetation that adorned Mount Sinai when the Torah was given. On a Kabbalistic level, vegetation alludes to the power of Torah study to protect us from harsh decrees.